A304531 -id:A304531 - cp's OEIS frontend

A304535 Restricted growth sequence transform of A278222(A304533(n)).

1, 2, 3, 2, 3, 2, 4, 2, 4, 5, 2, 4, 6, 4, 7, 3, 6, 4, 8, 4, 7, 9, 6, 10, 4, 6, 8, 5, 11, 6, 8, 2, 4, 6, 4, 6, 4, 5, 4, 5, 8, 2, 4, 6, 4, 6, 4, 6, 3, 6, 11, 3, 6, 12, 6, 9, 7, 9, 7, 10, 5, 11, 13, 11, 14, 6, 12, 15, 11, 13, 12, 16, 4, 5, 8, 5, 8, 5, 8, 6, 17, 5, 18, 5, 11, 19, 18, 20, 11, 19, 5, 8, 16, 8, 17, 2, 4, 5, 4, 5, 4, 5, 4, 5, 18, 4

Offset: 0

Antti Karttunen, May 15 2018

nonn

For all i, j: a(i) = a(j) => A304536(i) = A304536(j).

Antti Karttunen, Table of n, a(n) for n = 0..74430

Cf. A278222, A302791, A304531, A304533, A304536.

A304755 Suspected divisor-or-multiple permutation: a(1) = 1, and for n > 1, a(n) is either the second smallest divisor of a(n-1) not already present in sequence, or the smallest divisor if it is the only one not yet used, or (if all divisors have been already encountered), a(n) = a(n-1) * {the least power of the least prime not dividing a(n-1) such that the term is not already present}.

Original entry on oeis.org

1, 2, 6, 3, 12, 4, 36, 18, 9, 72, 24, 8, 216, 54, 27, 108, 540, 10, 5, 20, 60, 30, 15, 120, 40, 360, 90, 45, 180, 1260, 14, 7, 28, 84, 42, 21, 168, 56, 504, 126, 63, 252, 6300, 35, 70, 210, 105, 420, 140, 3780, 189, 378, 1890, 270, 135, 1080, 7560, 315, 630, 6930, 22, 11, 44, 132, 66, 33, 264, 88, 792, 198, 99, 396, 1980, 110, 55

Offset: 1

Antti Karttunen, May 20 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences that are permutations of the natural numbers

Cf. A304756 (inverse).
Cf. A053669, A304757.
Cf. A303751, also A282291, A304531 for variants.

up_to = 2^16;
A053669(n) = forprime(p=2, , if (n % p, return(p))); \\ From A053669
v304755 = vector(up_to);
m304756 = Map();
find_kth_unused_divisor(k,n,m_inverses) = { my(pd=0); fordiv(n,d,if(!mapisdefined(m_inverses,d),pd=d;k--); if((!k || (d == n)), return(pd))); };
prev=1; for(n=1,up_to, if((try = find_kth_unused_divisor(2,prev,m304756))!=0, mapput(m304756,v304755[n] = try,n), p = A053669(prev); while(mapisdefined(m304756,prev), prev *= p); v304755[n] = prev; mapput(m304756,prev,n)); prev = v304755[n]);
A304755(n) = v304755[n];
A304756(n) = mapget(m304756,n);

A304537 Suspected divisor-or-multiple permutation of squarefree numbers: a(n) = A019565(A304533(n)).

Original entry on oeis.org

1, 2, 6, 3, 15, 5, 65, 13, 26, 182, 7, 14, 42, 21, 105, 35, 455, 91, 910, 10, 30, 210, 70, 2730, 39, 78, 546, 273, 1365, 195, 7995, 41, 82, 246, 123, 615, 205, 2665, 533, 1066, 11726, 11, 22, 66, 33, 165, 55, 715, 143, 286, 2002, 77, 154, 462, 231, 1155, 385, 5005, 1001, 10010, 110, 330, 2310, 770, 30030, 429, 858, 6006, 3003, 15015, 2145, 87945, 451, 902

Offset: 0

Antti Karttunen, May 15 2018

nonn

Each a(n) is always either a divisor or a multiple of a(n+1).
Consider A052330. Imagine that it is an automatic piano that "plays sequences" when an appropriate punched "tape" is fed to it (as its input), i.e., when it is composed from the right with an appropriate sequence p, as A019565(p(n)). The 1-bits in the binary expansion of each p(n) are the "holes" in the tape, and they determine which "tunes" are present on beat n. The "tunes" are actually "Fermi-Dirac primes" (A050376) that are multiplied together.
If the tape is constructed in such a way that between the successive beats (when moving from p(n) to p(n+1)), either a subset of 0-bits are toggled on (changed to 1's), or a subset of 1-bits are toggled off (changed to 0's), but no both kind of changes occur simultaneously, then when fed as an input to this piano, the resulting sequence "s" (the output) is guaranteed to satisfy the condition that s(n+1) is either a multiple or a divisor of s(n). Furthermore, if the given sequence p is itself a permutation of natural numbers, then also the produced sequence is. For example, Gray code A003188 and its inverse A006068 are such sequences, and when given as an "input tape" for A052330, they produce permutations A207901 and A302783.
There is a simpler instrument, called "squarefree piano" (A019565), with which it is possible to produce similar divisor-or-multiple sequences, but that contain only squarefree numbers. Given A003188 or A006068 as an input tape for it produces correspondingly sequences A302033 and A284003.
This sequence is obtained by playing "squarefree piano" with the same tape which yields A304531 when "Fermi-Dirac piano" is played with it. However, in this case the sequence A304531 is produced by a greedy algorithm, and thus its tape (A304533) is actually a back-formation, obtained from the "music" (A304531) by applying "tape-recorder" (A052331) to it. Note that this in not a subsequence of A304531, as the terms occur in different order than the squarefree terms of A304531.
See also Peter Munn's Apr 11 2018 message on SeqFan-mailing list.

Antti Karttunen, Table of n, a(n) for n = 0..8447
Michel Marcus, Peter Munn, et al, Discussion on SeqFan-list, April 2018

Cf. A019565, A304531, A304533.

Cf. also A303760, A303771.

A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A304533(n) = A052331(A304531(1+n));
A304537(n) = A019565(A304533(n));

a(n) = A019565(A304533(n)) = A019565(A052331(A304531(1+n))).

A304534 Inverse of A304533.

Original entry on oeis.org

0, 1, 3, 2, 5, 19, 4, 20, 10, 11, 13, 12, 15, 22, 14, 21, 41, 42, 44, 43, 46, 60, 45, 61, 51, 52, 54, 53, 56, 63, 55, 62, 7, 8, 24, 25, 6, 81, 29, 88, 17, 9, 27, 26, 16, 18, 28, 23, 48, 49, 65, 66, 47, 157, 70, 301, 58, 50, 68, 67, 57, 59, 69, 64, 236, 237, 239, 238, 241, 255, 240, 256, 246, 247, 249, 248, 251, 258, 250, 257, 277, 278, 280, 279, 282, 296, 281

Offset: 0

Antti Karttunen, May 14 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10116
Index entries for sequences that are permutations of the natural numbers

Cf. A304533 (inverse).

Cf. A052330, A304531, A304532.

For all n >= 0, a(A304533(n)) = n.

a(n) = A304532(A052330(n))-1. [This formula works if A304531 and thus A304533 are indeed permutations, containing all natural numbers.]

cp's OEIS Frontend

A304535 Restricted growth sequence transform of A278222(A304533(n)).

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A304537 Suspected divisor-or-multiple permutation of squarefree numbers: a(n) = A019565(A304533(n)).

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A304534 Inverse of A304533.

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